2 solution.pdf - STAT 24400 Homework#2 Solution Stephen...

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STAT 24400: Homework #2 Solution Stephen Stigler 1
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Stephen Stigler STAT 24400 (Winter 2017): Homework #2 Solution Acknowledge : This solution is written by Youngseok Kim. Problem 1 (i) By definition of cdf, we have F p x q “ 0 , x 0 F p x q “ ª x 0 30 t 4 p 1 ´ t q dt 6 x 5 ´ 5 x 6 , 0 § x 1 F p x q “ 1 , x 1 Therefore, Pr p 0 . 2 § X § 0 . 5 q “ F p 0 . 5 q ´ F p 0 . 2 q “ 0 . 1078 The expectation of X is E p X q “ ª 1 0 30 x 5 p 1 ´ x q dx 0 . 7143 The median is obtained by solving F p x q “ 0 . 5 ñ 6 x 5 ´ 5 x 6 0 . 5 ñ x 0 . 7356 Note that Y “ p X ` 2 q 1 { 2 is an increasing function. F Y p y q “ Pr p Y § y q “ Pr p X § y 2 ´ 2 q “ F X p y 2 ´ 2 q Di erentiating both hands, we have f Y p y q “ 2 yf X p y 2 ´ 2 q Therefore we have f Y p y q “ 60 y p y 2 ´ 2 q 4 p 3 ´ y 2 q , ? 2 § y § ? 3 and otherwise 0. (ii) F p x q “ 0 , x 0 . 2 F p x q “ ª x 0 . 2 e ´p t ´ 0 . 2 q dt 1 ´ e ´p x ´ 0 . 2 q , x § 0 . 2 Pr p 0 . 2 § X § 0 . 5 q “ F p 0 . 5 q ´ F p 0 . 2 q “ 0 . 2592 E p X q “ ª 8 0 . 2 xe ´p x ´ 0 . 2 q dx 0 . 8931 F p x q “ 0 . 5 ñ 1 ´ e ´p x ´ 0 . 2 q 0 . 5 ñ x 0 . 7356 f Y p y q “ 2 yf X p y 2 ´ 2 q “ 2 ye ´p y 2 ´ 2 . 2 q , y ? 2 . 2 and otherwise 0. Problem 1 continued on next page. . . Page 2 of 6
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Stephen Stigler STAT 24400 (Winter 2017): Homework #2 Solution (iii) F p x q “ 0 , x 0 F p x q “ ª x 0 1 2 sin tdt 1 2 p 1 ´ cos x q , 0 § x F p x q “ 1 , x § Pr p 0 . 2 § X § 0 . 5 q “ F p 0 . 5
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